In assum3, the absolute compression error over the unit ball is bounded by \(\delta \) . Since we do not impose any condition on \(\mathsf {Q}(\mathbf {x})\)  for \(\Vert \mathbf {x}\Vert _{p}>1\) , assum3 is strictly weaker than the deterministic version of assum1, and covers the compressors in [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. To our best knowledge, Moniqua [10]} is the only distributed method that converges sublinearly under a special case \(p=\infty \)  of assum3. In this work, we design a dynamical scaling mechanism to achieve linear convergence in sec4b.
